Problem: What is the inverse of the function $f(x)=8x+1$ ? $f^{-1}(x)=$
Solution: Let's start by replacing $f(x)$ with $y$. $y=8x+1$ If a function contains the point $(a,b)$, the inverse of that function contains the point $(b,a)$. So if we swap the position of $x$ and $y$ in the equation, we get the inverse relationship. In this case, the function is $y=8x+1$, so the inverse relationship is $x=8y+1$. Solving this equation for $y$ will give us an expression for $f^{-1}(x)$. $\begin{aligned} x&=8y+1\\\\ x-1&=8y\\\\ \dfrac{x-1}{8}&=y\\\\\\ \end{aligned}$ The inverse of the function is $f^{-1}(x)=\dfrac{x-1}{8}$. [I saw someone solve this problem by originally solving for x. Were they wrong?]